Now it is a remarkable fact [1]}, [2]}, [3]}, [4]}, [5]}, [6]} that if we add photons to the interacting theory then one can add correction terms to \(\tilde{Q}_0(f)\)  and the corrected charge \(Q(f)\)  commutes with the \(S\) -matrix. In this case the total charge can be written as, \(Q(f) = Q_H(f) + Q_S(f)\) . \(Q_H(f)\)  is called the hard-charge and is required to generate the \(U(1)\)  transformation on the charged particles, i.e,
\({\begin{array}{c}Q^{in}_{H}(f) {\alpha , in} = \bigg ( \sum _{k\in \alpha } q_k f(x_k)\bigg ) {\alpha , in} \\{\beta , out} Q^{out}_{H}(f) = {\beta , out} \bigg ( \sum _{k\in \beta } q_k f(x_k)\bigg )\end{array}}\) 
